attachments_22-12-2011_21-07-09 / razdel 4

y8 = f 8(x0 ) =

lim

6y =

lim

f ( x0 + 6x) f ( x0 )

.

6x 0

6x

6x 0

6x

r

&

a

r

e1

,e2

a

r r

r

a = xe1 + ye2 , x,y- a .

a x + a

x

+ ... + a

x

= b ,

5

11 1

12

2

1n

n

1

5 a21x1 + a22x2 + ... + a2nxn = b2,

................

.

5

5a

x + a

x

+ ... + a

x

= b .

m1 1

m2

2

mn n

m

M n ( ( n

a11x1 + a12x2 + ,,, + a1nxn = b1,

5

5a21x1 + a22x2 + ,,, + a2nxn = b2,

.............

.

5

5a x

+ a x

+ ,,, + a x

= b .

n1 1

n2 2

nn

n

n

# n ( ( n

a

x

+ a

x

+ ... + a

x

= 0,

5 11 1

12

2

1n

n

5a21x1 + a22x2 + ... + a2nxn = 0,

................

.

5

5a

x

+ a

x

+ ... + a

x

= 0.

n1 1

n2

2

nn n

# a b

,

r

r

r

r

r

& a ,b

(a,b) : a , b

=| a |

,| b |,cos( a=b ).

# ! :

f (a + 0) f (a 0) = 6a f

! f (x)

! a .

r r

# ' ( (

a,b, c

, a

b ×

r

c .

$ .

H ! $ xk (k = 1, 2,..., n)

&,

( D ,

Dk -

, " (

(

k -

& & & .

? ! ! + :

( (

f (a 0) ,

f (a + 0) &,

a

!

! +

.

a -

? !

! + :

1- ,

f (a 0) f (a + 0) , a ! ! + .

? ! : a

!

f ( x) , ! (

f ( x)

. E ,

f (a 0) = f (a + 0) = f (a) .

? ! 1-+ :

a

f (x)

! 1-+

,

&

f (a 0)

f (a + 0) " .

a

f (x)

? ! 2-+ :

! 2-+ , ,

( ( ,

f (a 0) ,

f (a + 0) &

".

a -

? ! +

:

1-

,

a11

a12

...

a1n

a11

a21 ...

an1

A =

a21

a22

...

a2n

AT =

a12

a22 ...

a2n

.

... ...

...

...

... ... ... ...

an1 an2

...

ann

a1n

a2n ...

ann

E+ ! , ( !

, .

Dn

x

=

D1

,

x

=

D2

,

...,

x

=

( L).

D

M <: 1

D

2

D

n

M:

x X ( X -

" ),

y Y ; , X

f $

y = f ( x), x X .

M !

':

f (x)

!

' a ( x

a ), lim f (x) = .

x

a

a ( x

a ), lim f (x) = 0.

x

a

y = f (x)

M *:

*

X ,

&

x1, x2 X

x1 < x2

, f (x1 ) < f (x2 ) .

M – " & ".

f ( x)

M !:

( a .

@

f ( x) (

a , x

a

a , . .

lim f ( x) = f (a).

x a

f ( x)

M

:

(a,b)

f (a + 0) = f (a) , f (b 0) = f (b).

M

:

f ( x)

( X

(

&), ( ! .

M

!: y = f (x)

&

X

B (,

&

x X

M !:

y = f (x) &

X

(, " T > 0 , :

1)

x X , x T X

x + T X ;

2)

& x X

f (x) = f (x + T ) ;

3)

T $ .

E $

T

y = f (x) .

M :

y = f (u) , u U ,

u = ( x), x X ,

y(x) = f (u(x)) ( (.

M *: y = f (x) & " (

X ,

& x1, x2 X

x2

> x1 ,

f (x2 ) < f (x1) .

M !: y = f (x)

& X

,

lim +(x)

= 1.

x a 3(x)

( :

x2

+

y2

= 1

, a b - & .

a2

b2

-

, (

(

x2

+

y2

= z , a b -

a2

b2

1

0

1

% 1. 1) H

2

1

0

,

0

1

2

! ( .

& '.

1

0

1

= 1, ( 1)1+1

1

0

+ 0 ,

( 1)1+2

2

0

+ ( 1), ( 1)1+3

2

1

=

2

1

0

1 2

0 2

0 1

0

1

2

= 1,1(1,

2 0( 1)) + 0 , ( 1)(4 0)+ ( 1),1( 2 0) = 2 + 0 + 2 = 4.

3

2

1

2) H

4

2

1

, !

0

3

1

&.